Question 1

Complete the sentence so the result is an identity. Let x be any real number.
-Write $\sec x$ in terms of $\sin x$.

Accepted Answer

A)  $\pm \sqrt { 1 - \sin ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
C)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { \sin x }$ 
D)  $\frac { \pm \sin x \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
A)  $\pm \sqrt { 1 - \sin ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$ 
C)  $\frac { \pm \sqrt { 1 - \sin ^ { 2 } x } } { \sin x }$ 
D)  $\frac { \pm \sin x \sqrt { 1 - \sin ^ { 2 } x } } { 1 - \sin ^ { 2 } x }$

Question 2

Use Identities to find the exact value.
-$\cot 5 \theta = \tan 4 \theta$

Accepted Answer

A)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = 40 ^ { \circ }$ 
D)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
A)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = 40 ^ { \circ }$ 
D)  $\theta = \frac { 17 ^ { \circ } } { 7 }$

Question 3

Using a sum or difference identity, write the following as an expression involving functions of x.
-Find $\sin \frac { \theta } { 2 }$, given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$.

Accepted Answer

A)  $- \frac { \sqrt { 30 } } { 10 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 5 } } { 5 }$ 
A)  $- \frac { \sqrt { 30 } } { 10 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 5 } } { 5 }$

Question 4

Use identities to find the indicated value for each angle measure.
-$$\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  $- 1$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 2 } } { 2 }$ 
C)  $- 1$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 5

Use a sum or difference identity to find the exact value.
-$$\frac { \sec \theta \sin \theta } { \tan \theta } - 1$$

Accepted Answer

A)  0 
B)  $- \sec ^ { 2 } \theta$ 
C)  1 
D)  $1 - \sin \theta$ 
A)  0 
B)  $- \sec ^ { 2 } \theta$ 
C)  1 
D)  $1 - \sin \theta$

Question 6

Write in terms of the cofunction of a complementary angle.
-$\sin \frac { \pi } { 17 }$

Accepted Answer

A)  $\csc \frac { 15 \pi } { 34 }$ 
B)  $\csc \frac { 16 \pi } { 17 }$ 
C)  $\cos \frac { 16 \pi } { 17 }$ 
D)  $\cos \frac { 15 \pi } { 34 }$ 
A)  $\csc \frac { 15 \pi } { 34 }$ 
B)  $\csc \frac { 16 \pi } { 17 }$ 
C)  $\cos \frac { 16 \pi } { 17 }$ 
D)  $\cos \frac { 15 \pi } { 34 }$

Question 7

Verify that each equation is an identity.
-$$7 \csc ^ { 2 } \theta - 3 \cot ^ { 2 } \theta = 4 \csc ^ { 2 } \theta + 3$$

Accepted Answer

\[\begin{array} { l } 
7 \csc ^ { 2 } \t

Question 8

Find the exact value of the expression using the provided information.
-$\cos 165 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
A)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$

Question 9

Verify that each equation is an identity.
-$$\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\sec...

Question 10

Perform the transformation.
-$$\cot 65 ^ { \circ } = \frac { 1 } { ---^{25^ { \circ }} }$$

Accepted Answer

A)  $\csc$ 
B)  $\sec$ 
C)  $\tan$ 
D)  $\cot$ 
A)  $\csc$ 
B)  $\sec$ 
C)  $\tan$ 
D)  $\cot$

Question 11

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\tan \left( x - 360 ^ { \circ } \right)$

Accepted Answer

A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $\tan x$ 
C)  $- \tan x$ 
D)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $\tan x$ 
C)  $- \tan x$ 
D)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$

Question 12

Verify that the equation is an identity.
-$$\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alpha$$

Accepted Answer

\[\begin{array} { l } 
\frac { \sin ( \a

Question 13

Solve the problem.
-$\sin \frac { 5 \pi } { 12 }$

Accepted Answer

A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question 14

Verify that the equation is an identity.
-$$\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alpha$$

Accepted Answer

\[\begin{array} { l } 
\frac { \cos ( \a

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Suppose that for an electrical appliance, voltage is given by $\mathrm { V } = 189 \sin 120 \pi \mathrm { t }$ and amperage by $\mathrm { I } = 1.07 \sin 120 \pi \mathrm { t }$, where $\mathrm { t }$ is time in seconds. The wattage is given by $\mathrm { W } = \mathrm { VI }$. Find the maximum wattage used by the appliance.

Accepted Answer

A)  $101.115$ 
B)  189 
C)  $404.46$ 
D)  $202.23$ 
A)  $101.115$ 
B)  189 
C)  $404.46$ 
D)  $202.23$

Question 16

Use Identities to find the exact value.
-$\sin \left( x + 45 ^ { \circ } \right)$

Accepted Answer

A)  $\sin x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
A)  $\sin x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$

Question 17

Decide whether the expression is or is not an identity.
-Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta < 0$.

Accepted Answer

A)  $\frac { \sqrt { 5 } } { 5 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { \sqrt { 5 } } { 2 }$ 
D)  $- 5$ 
A)  $\frac { \sqrt { 5 } } { 5 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { \sqrt { 5 } } { 2 }$ 
D)  $- 5$

Question 18

Verify that each equation is an identity.
-$$( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[(...

Question 19

Verify that the equation is an identity.
-$$\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x$$

Accepted Answer

\[\begin{array} { l } 
\left( \cos \frac

Question 20

Verify that the equation is an identity.
-$$\csc ( \mathrm { s } + \mathrm { t } ) = \frac { \csc \csc t } { \cot \mathrm { t } + \cot \mathrm { s } }$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\csc...

Complete the sentence so the result is an identity. Let x be any real number. -Write $\sec x$ in terms of $\sin x$ .

Use Identities to find the exact value. - $\cot 5 \theta = \tan 4 \theta$

Using a sum or difference identity, write the following as an expression involving functions of x. -Find $\sin \frac { \theta } { 2 }$ , given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$ .

Use identities to find the indicated value for each angle measure. - $\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$

Use a sum or difference identity to find the exact value. - $\frac { \sec \theta \sin \theta } { \tan \theta } - 1$

Write in terms of the cofunction of a complementary angle. - $\sin \frac { \pi } { 17 }$

Verify that each equation is an identity. - $7 \csc ^ { 2 } \theta - 3 \cot ^ { 2 } \theta = 4 \csc ^ { 2 } \theta + 3$

Find the exact value of the expression using the provided information. - $\cos 165 ^ { \circ }$

Verify that each equation is an identity. - $\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }$

Perform the transformation. - $\cot 65 ^ { \circ } = \frac { 1 } { ---^{25^ { \circ }} }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\tan \left( x - 360 ^ { \circ } \right)$

Verify that the equation is an identity. - $\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alpha$

Solve the problem. - $\sin \frac { 5 \pi } { 12 }$

Verify that the equation is an identity. - $\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alpha$

Use Identities to find the exact value. - $\sin \left( x + 45 ^ { \circ } \right)$

Decide whether the expression is or is not an identity. -Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta < 0$ .

Verify that each equation is an identity. - $( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1$

Verify that the equation is an identity. - $\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x$

Verify that the equation is an identity. - $\csc ( \mathrm { s } + \mathrm { t } ) = \frac { \csc \csc t } { \cot \mathrm { t } + \cot \mathrm { s } }$

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Complete the sentence so the result is an identity. Let x be any real number. -Write sec⁡x\sec xsecx in terms of sin⁡x\sin xsinx .

Use Identities to find the exact value. - cot⁡5θ=tan⁡4θ\cot 5 \theta = \tan 4 \thetacot5θ=tan4θ

Use identities to find the indicated value for each angle measure. - cos⁡7π12cos⁡5π12+sin⁡7π12sin⁡5π12\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }cos127π​cos125π​+sin127π​sin125π​

Use a sum or difference identity to find the exact value. - sec⁡θsin⁡θtan⁡θ−1\frac { \sec \theta \sin \theta } { \tan \theta } - 1tanθsecθsinθ​−1

Write in terms of the cofunction of a complementary angle. - sin⁡π17\sin \frac { \pi } { 17 }sin17π​

Verify that each equation is an identity. - 7csc⁡2θ−3cot⁡2θ=4csc⁡2θ+37 \csc ^ { 2 } \theta - 3 \cot ^ { 2 } \theta = 4 \csc ^ { 2 } \theta + 37csc2θ−3cot2θ=4csc2θ+3

Find the exact value of the expression using the provided information. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

Verify that each equation is an identity. - sec⁡(2θ)=csc⁡2θcsc⁡2θ−2\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }sec(2θ)=csc2θ−2csc2θ​

Perform the transformation. - cot⁡65∘=1−−−25∘\cot 65 ^ { \circ } = \frac { 1 } { ---^{25^ { \circ }} }cot65∘=−−−25∘1​

Use an identity to write the expression as a single trigonometric function or as a single number. - tan⁡(x−360∘)\tan \left( x - 360 ^ { \circ } \right)tan(x−360∘)

Verify that the equation is an identity. - sin⁡(α−β)sin⁡αsin⁡β=cot⁡β−cot⁡α\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alphasinαsinβsin(α−β)​=cotβ−cotα

Solve the problem. - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​

Verify that the equation is an identity. - cos⁡(α+β)cos⁡αsin⁡β=cot⁡β−tan⁡α\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta } = \cot \beta - \tan \alphacosαsinβcos(α+β)​=cotβ−tanα

Use Identities to find the exact value. - sin⁡(x+45∘)\sin \left( x + 45 ^ { \circ } \right)sin(x+45∘)

Decide whether the expression is or is not an identity. -Find sin⁡θ\sin \thetasinθ if cot⁡θ=−2\cot \theta = - 2cotθ=−2 and cos⁡θ<0\cos \theta < 0cosθ<0 .

Verify that each equation is an identity. - (sec⁡α−tan⁡α)(sec⁡α+tan⁡α)=1( \sec \alpha - \tan \alpha ) ( \sec \alpha + \tan \alpha ) = 1(secα−tanα)(secα+tanα)=1

Verify that the equation is an identity. - (cos⁡x2−sin⁡x2)2=1−sin⁡x\left( \cos \frac { x } { 2 } - \sin \frac { x } { 2 } \right) ^ { 2 } = 1 - \sin x(cos2x​−sin2x​)2=1−sinx

Verify that the equation is an identity. - csc⁡(s+t)=csc⁡csc⁡tcot⁡t+cot⁡s\csc ( \mathrm { s } + \mathrm { t } ) = \frac { \csc \csc t } { \cot \mathrm { t } + \cot \mathrm { s } }csc(s+t)=cott+cotscsccsct​